Amsco's AP Calculus AB BC: Preparing for the Advanced Placement Exams

Maxine Lifshitz, Martha Green

Chapter 5 Applications of the Derivative - all with Video Answers

Educators

Section 1

Three Theorems: The Extreme Value Theorem, Rolle’s Theorem, and the Mean Value Theorem

01:06

Problem 1

A graphing calculator is required for some questions.
Which of the following functions satisfy the hypotheses of Rolle's Theorem on the interval $[0,2]$ ?
I $f(x)=\frac{1}{|x-1|}$
II $f(x)=|x-1|$
III $f(x)=x^2-2 x$
(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

Colin Cassell

Numerade Educator

01:12

Problem 2

A graphing calculator is required for some questions.
Which of the following functions satisfy the hypotheses of the Mean Value Theorem on the interval $[0,2]$ ?
I $f(x)=\sin \pi x+\cos 2 x$
II $f(x)=\sqrt[3]{x-8}$
III $f(x)=\left|x^2-2 x\right|$
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II, and III

Rikhil Makwana

Numerade Educator

01:22

Problem 3

A graphing calculator is required for some questions.
At what value(s) of $x$ in the interval $[0,2]$ is the slope of the line tangent to $f(x)=x^3-3 x^2$ equal to the slope of the line joining $(0, f(0))$ and $(2, f(2))$ ?
(A) 0
(B) 0.423
(C) 1.577
(D) 2
(E) 0.423 and 1.577

Chris Bolognese

Numerade Educator

01:18

Problem 4

A graphing calculator is required for some questions.
Find the absolute minimum value of $f(x)=x^2-3 x^4$ on the interval $[-1,1]$.
(A) -2
(B) 0
(C) 0.408
(D) 1
(E) There is no absolute minimum on the interval.

Carson Merrill

Numerade Educator

00:26

Problem 5

A graphing calculator is required for some questions.
How many values of $c$ satisfy the conclusion of the Mean Value Theorem for $f(x)=x^3+1$ on the interval $[-1,1]$ ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Amrita Bhasin

Numerade Educator

00:52

Problem 6

A graphing calculator is required for some questions.
How many values of $c$ are guaranteed by Rolle's Theorem for the function $f(x)=\frac{\sin x}{x}$ on the interval $[-10,10]$ ?
(A) 4
(B) 5
(C) 6
(D) 7
(E) The theorem does not apply.

Amrita Bhasin

Numerade Educator